Chaotic functions with zero topological entropy
Author:
J. Smítal
Journal:
Trans. Amer. Math. Soc. 297 (1986), 269282
MSC:
Primary 58F13; Secondary 54H20, 58F11
MathSciNet review:
849479
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Abstract: Recently Li and Yorke introduced the notion of chaos for mappings from the class , where is a compact real interval. In the present paper we give a characterization of the class of mappings chaotic in this sense. As is well known, contains the mappings of positive topological entropy. We show that contains also certain (but not all) mappings that have both zero topological entropy and infinite attractors. Moreover, we show that the complement of consists of maps that have only trajectories approximate by cycles. Finally, it turns out that the original Li and Yorke notion of chaos can be replaced by (an equivalent notion of) chaos, distinguishable on a certain level .
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 [1]
 L. Block, Homoclinic points of mappings of the interval, Proc. Amer. Math. Soc. 72 (1978), 576580. MR 509258 (81m:58063)
 [2]
 , Stability of periodic orbits in the theorem of Šarkovskii, Proc. Amer. Math. Soc. 82 (1981), 333336. MR 593484 (82b:58071)
 [3]
 G. J. Butler and G. Pianigiani, Periodic points and chaotic functions in the unit interval, Bull. Austral. Math. Soc. 18 (1978), 255265. MR 0494303 (58:13203)
 [4]
 W. A. Coppel, Maps of an interval, Preprint, Univ. of Minnesota, 1983.
 [5]
 R. Graw, On the connection between periodicity and chaos of continuous functions and their iterates, Aequationes Math. 19 (1979), 277278.
 [6]
 J. Harrison, Wandering intervals, Dynamical Systems and Turbulence (Warwick, 1980), Lecture Notes in Math., vol. 898, Springer, Berlin, Heidelberg and New York, 1981, pp. 154163. MR 654888 (83h:58059)
 [7]
 I. Kan, A chaotic function possessing a scrambled set with positive Lebesgue measure, Proc. Amer. Math. Soc. 92 (1984), 4549. MR 749887 (86b:26009a)
 [8]
 Ch. K. Kenžegulov and A. N. Šarkovskii, On properties of the set of limit points of an iterated sequence of a continuous function, Volž. Mat. Sb. 3 (1965), 343348. (Russian) MR 0199316 (33:7464)
 [9]
 L. Kuipers and L. Niederreiter, Uniform distribution of sequences, Wiley, New York, 1974, p. 8. MR 0419394 (54:7415)
 [10]
 C. Kuratowski, Topologie. I, PWN, Warsaw, 1958.
 [11]
 T. Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), 985992. MR 0385028 (52:5898)
 [12]
 M. Misiurewicz, Horseshoes for mappings of the interval, Bull. Acad. Polon. Sci. Sér. Math. 27 (1979), 167169. MR 542778 (81b:58033)
 [13]
 , Chaos almost everywhere, Preprint, 1983.
 [14]
 A. N. Šarkovskii, Coexistence of cycles of continuous maps of the line into itself, Ukrain. Mat. Ž. 16 (1964), 6171. (Russian) MR 0159905 (28:3121)
 [15]
 , On cycles and the structure of continuous mappings, Ukrain. Mat. Ž. 17 (1965), 104111. (Russian) MR 0186757 (32:4213)
 [16]
 , The behavior of a map in a neighborhood of an attracting set, Ukrain. Mat. Ž. 18 (1966), 6083. (Russian) MR 0212784 (35:3649)
 [17]
 , The partially ordered system of attracting sets, Soviet Math. Dokl. 7 (1966), 13841386.
 [18]
 , Attracting sets containing no cycles, Ukrain. Mat. Ž. 20 (1968), 136142. (Russian) MR 0225314 (37:908)
 [19]
 , On some properties of discrete dynamical systems, Proc. Internat. Colloq. on Iteration Theory and its Applications, Toulouse, 1982.
 [20]
 J. Smítal and K. Smítalová, Structural stability of nonchaotic difference equations, J. Math. Anal. Appl. 90 (1982), 111; Errata 101 (1984), p. 324. MR 746238 (85f:58065)
 [21]
 J. Smital, A chaotic function with some extremal properties, Proc. Amer. Math. Soc. 87 (1983), 5456. MR 677230 (84h:26008)
 [22]
 , A chaotic function with a scrambled set of positive Lebesgue measure, Proc. Amer. Math. Soc. 92 (1984), 5054. MR 749888 (86b:26009b)
 [23]
 J. Smital and K. Janková, Characterization of chaos, Bull. Austral. Math. Soc. (to appear). MR 854575 (87k:58178)
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 P. Štefan, A theorem of Šarkovskii on existence of periodic orbits of a continuous endomorphisms of the real line, Comm. Math. Phys. 54 (1977), 237248. MR 0445556 (56:3894)
 [25]
 M. B. Verejkina and A. N. Šarkovskii, Recurrence in onedimensional dynamical systems, Approx. and Qualitative Methods of the Theory of DifferentialFunctional Equations, Inst. Math. Akad. Nauk USSR, Kiev, 1983, pp. 3546. (Russian) MR 753681 (85m:58149)
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DOI:
http://dx.doi.org/10.1090/S00029947198608494799
PII:
S 00029947(1986)08494799
Article copyright:
© Copyright 1986
American Mathematical Society
